If R = {(x, y) : x and y are integers and x^2 + y^2 = 64} is a relation, then find R.

The function is continuous in the domain x ≥ 3/4

Here we have a root function:

f(x) = ⁴√(4x - 3)

This is an even degree root function, so we have problems when the argument is negative.

Then the allowed values (where the function is defined, and thus, continuous) are:

4x - 3 ≥ 0

4x ≥ 3

x ≥ 3/4

There the function is continuous.

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The sentence that best describes the paragraph is: "The pirates didn't believe Caesar would kill them because he was their prisoner." So, the correct option is c) Pirates didn’t believe Caesar will kill them because he was their prisoner.

The paragraph recounts the events of Caesar's kidnapping by pirates while crossing the Aegean Sea. Despite being held captive, Caesar maintained an attitude of superiority. The pirates demanded a ransom of 20 talents of silver, but Caesar insisted they ask for 50. This indicates that Caesar saw himself as more valuable than the initial ransom amount suggested by the pirates.

After the ransom was paid, Caesar did not forget the pirates' actions. He raised a fleet, pursued the pirates, and captured them. The sentence implies that the pirates didn't believe Caesar would actually kill them because he was their prisoner and they likely saw his promises as mere jest.

However, Caesar, true to his word, had the pirates crucified on his own authority.

This sequence of events highlights Caesar's determination, strategic thinking, and his ability to command respect even in dire circumstances. Despite being initially taken prisoner, he managed to turn the tables on the pirates, assert his authority, and exact his revenge. So, the correct option is c) Pirates didn’t believe Caesar will kill them because he was their prisoner.

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a. An equation to describe the total cost of RV rental:

Cost = (125 * d) + (0.32 * m) + 2500

b. Comparing the two costs will determine which option is cheaper.

c. For the more direct route: m ≤ 3500

For fewer overnight stays: d ≤ 14

These domains ensure that we don't exceed the original values for miles and days.

d. I recommend compromising by lessening the number of days of the rental. By reducing the rental period to 11 days, you can stay within the $5,000 budget while still allowing your little sister to take the two-week trip.

a. To write an equation for the total cost of RV rental, we can use the given information. The cost per day is $125, and there is an additional charge of 32 cents per mile. Let's denote the number of days as d and the number of miles as m. The equation for the total cost of RV rental can be written as:

Cost = (125 * d) + (0.32 * m) + 2500

b. To compare the costs of the two options, we need to calculate the total cost for each. Option 1 has 3500 miles and takes 11 days, while option 2 has 3000 miles and takes 14 days. We can substitute these values into the equation from part a to find the total costs for each option. Comparing the two costs will determine which option is cheaper.

c. To compromise and stay within a budget of $5,000, we can adjust either the number of miles or the number of days. For the more direct route, we can reduce the number of miles, and for fewer overnight stays, we can reduce the number of days. The domains for these compromises would be:

For the more direct route: m ≤ 3500

For fewer overnight stays: d ≤ 14

These domains ensure that we don't exceed the original values for miles and days.

d. To find the number of miles or days to eliminate in order to stay under the $5,000 budget, we can set up equations using the total cost equation from part a. Let's denote the reduced number of miles as m' and the reduced number of days as d'. We need to solve the following equation for each compromise:

(125 * d') + (0.32 * m') + 2500 ≤ 5000

By substituting the appropriate values into the equation and solving for m' or d', we can determine how many miles or days need to be eliminated.

Based on the given information, I recommend compromising by lessening the number of days of the rental. By reducing the rental period to 11 days, you can stay within the $5,000 budget while still allowing your little sister to take the two-week trip. This compromise ensures that you don't have to sacrifice too much scenic beauty or make drastic changes to the route.

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The pairs of angles that are vertical angles in the figure are R and T.

In the figure provided, vertical angles are formed by the intersection of two lines. Vertical angles are always congruent (equal in measure) to each other.

Looking at the given options:

R and T are vertical angles because they are formed by the intersection of lines.

P and D are not vertical angles. They are adjacent angles formed by the intersection of lines, but they are not directly opposite each other.

Therefore, the pairs of angles that are vertical angles in the figure are:

R and T

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Answer:

53)

[tex]f(x) = \frac{7(x - 4)(x + 6)}{(x + 4)(x + 5)} [/tex]

The equation of the ellipse with foci (7, 17) and (7, -13), and a major axis of length 34 is[tex](x^2/289) + (y^2/225) = 1.[/tex]

To determine the equation of an ellipse given its foci and the length of its major axis, we need to use the standard form equation for an ellipse. The standard form equation for an ellipse centered at the origin is:

[tex](x^2/a^2) + (y^2/b^2) = 1[/tex]

where 'a' represents the semi-major axis and 'b' represents the semi-minor axis.

In this case, we know that the distance between the foci is equal to 2a, which means a = 34/2 = 17. The foci of the ellipse are given as (7, 17) and (7, -13). The foci lie on the major axis of the ellipse, and since their y-coordinates differ by 30 (17 - (-13) = 30), the length of the major axis is equal to 2b, which means b = 30/2 = 15.

Now we have the values of a and b, so we can substitute them into the standard form equation:

[tex](x^2/17^2) + (y^2/15^2) = 1[/tex]

Simplifying further, we have:

[tex](x^2/289) + (y^2/225) = 1[/tex]

Therefore, the equation of the ellipse with foci (7, 17) and (7, -13), and a major axis of length 34 is:

[tex](x^2/289) + (y^2/225) = 1.[/tex]

This equation represents an ellipse centered at the point (0, 0) with a semi-major axis of length 17 and a semi-minor axis of length 15.

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Answer:

The equation of the ellipse with foci (7, 17) and (7, -13), and a major axis of length 34 is

To determine the equation of an ellipse given its foci and the length of its major axis, we need to use the standard form equation for an ellipse. The standard form equation for an ellipse centered at the origin is:

where 'a' represents the semi-major axis and 'b' represents the semi-minor axis.

In this case, we know that the distance between the foci is equal to 2a, which means a = 34/2 = 17. The foci of the ellipse are given as (7, 17) and (7, -13). The foci lie on the major axis of the ellipse, and since their y-coordinates differ by 30 (17 - (-13) = 30), the length of the major axis is equal to 2b, which means b = 30/2 = 15.

Now we have the values of a and b, so we can substitute them into the standard form equation:

Simplifying further, we have:

Therefore, the equation of the ellipse with foci (7, 17) and (7, -13), and a major axis of length 34 is:

This equation represents an ellipse centered at the point (0, 0) with a semi-major axis of length 17 and a semi-minor axis of length 15.

Answer: -15/17

Step-by-step explanation:

sin ∅ = 8/17

If you drew a a line in the 2rd quadrant because tan ∅ <0, which means tan∅ is negative.

Tan∅= sin∅/cos∅            

they told you sin∅ is positive which is related to your y.

but cos∅ needs to be negative which is related to x

This happens in the second quadrant x is negative and y is positive.

Now we know which way to draw our line.  Label the opposite of the angle 8 and the hypotenuse 17 because sin∅ = 8/17

Use pythagorean to find adjacent.

17² = 8² + a²

225 = a²

a = 15

The adjacent is negative because the adjacent is on the x-axis in the negative direction.

cos ∅ = adj/hyp

cos∅ =  -15/17

Answer: -15/17

Step-by-step explanation:

In the first quadrant, all sine, cosine and tan are all positive. In the second quadrant, only sine is positive. Third quadrant, only tangent is positive and fourth quadrant, only cosine is positive.

Therefore, when sine is positive and tan is negative, the angle can only be in quadrant 2. Then draw the triangle. Draw a triangle with the angle from the origin, with the opposite leg from the angle with value of 8 and the hypotenuse of value 17. Since cosine is what the question is asking for, and we know the data given forms an right triangle, the value of the other leg is 15. It is an 8-15-17 special triangle or use the Pythagorean Theorem.

Finally, taking the cosine of the angle from the origin gives -15/17 since it is in quadrant 2.

The functions y₁(t) = e^ãt cos(μt) and y₂(t) = e^ãt sin(μt) are a fundamental set of solutions because they are linearly independent and satisfy the given homogeneous linear differential equation, allowing for the formation of the general solution.

To show that y₁(t) = e^ãt cos(μt) and y₂(t) = e^ãt sin(μt) are a fundamental set of solutions, we need to demonstrate two things: linear independence and satisfaction of the given homogeneous linear differential equation.

First, let's consider linear independence. We can prove it by showing that there is no constant c₁ and c₂, not both zero, such that c₁y₁(t) + c₂y₂(t) = 0 for all t.

Now, let's verify that y₁(t) and y₂(t) satisfy the homogeneous linear differential equation. If the given differential equation is of the form ay''(t) + by'(t) + cy(t) = 0, we can substitute y₁(t) and y₂(t) into the equation and verify that it holds true.

Once we have established linear independence and satisfaction of the differential equation, we can state that the general solution to the homogeneous linear differential equation is given by y(t) = c₁y₁(t) + c₂y₂(t), where c₁ and c₂ are arbitrary constants. This general solution represents the linear combination of the fundamental set of solutions.

In summary, y₁(t) = e^ãt cos(μt) and y₂(t) = e^ãt sin(μt) form a fundamental set of solutions for the given differential equation, and the general solution is given by y(t) = c₁y₁(t) + c₂y₂(t).

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Answer:

$3606.44

Step-by-step explanation:

The question asks us to calculate the principal amount that needs to be invested in order to earn an interest of $6180 in 28 years at an annual interest rate of 6.12%.

To do this, we need to use the formula for simple interest:

[tex]\boxed{I = \frac{P \times R \times T}{100}}[/tex],

where:

I = interest earned

P = principal invested

R = annual interest rate

T = time

By substituting the known values into the formula above and then solving for P, we can calculate the amount that James needs to invest:

[tex]6180 = \frac{P \times 6.12 \times 28}{100}[/tex]

⇒ [tex]6180 \times 100 = P \times 171.36[/tex]     [Multiplying both sides by 100]

⇒ [tex]P = \frac{6180 \times 100}{171.36}[/tex]    [Dividing both sides of the equation by 171.36]

⇒ [tex]P = \bf 3606.44[/tex]

Therefore, James needs to invest $3606.44.

The missing statement 4 of the two column proof of AD ║ BC is:

Statement 4: m∠1 + m∠4 = 180°

The complete two column proof to show that AD || BC is as follows:

Statement 1: AD ║ DC

Reason 1: Given

Statement 2: m∠2 = m∠4

Reason 2: Given

Statement 3: ∠1 and ∠3 are supplements

Reason 3: Same side interior angles theorem

Statement 4: m∠1 + m∠4 = 180°

Reason 4: Def. of Supplementary angles

Statement 5: m∠1 + m∠2 = 180°

Reason 5: Substitution

Statement 6: ∠1 and ∠2 are supplements

Reason 6: Def. of Supplementary angles

Statement 7: AD ║ BC

Reason 7: Converse same side interior angles thm

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Answer:

Step-by-step explanation:

To find the maximum width of an aircraft seat that will accommodate 98% of all adult women, we need to determine the corresponding z-score for the 98th percentile of the normal distribution.

First, we find the z-score corresponding to the 98th percentile using a standard normal distribution table or calculator. The z-score for the 98th percentile is approximately 2.05.

Next, we use the z-score formula to find the corresponding value in the original distribution:

z = (x - μ) / σ

Solving for x (the maximum width of the aircraft seat):

x = z * σ + μ

Substituting the values given:

x = 2.05 * 4.36 + 37.6

x ≈ 45.98

Therefore, the maximum width of an aircraft seat that will accommodate 98% of all adult women is approximately 46 cm (rounded to one decimal place).

Answer:

56) f(x) = c(x - 1)²/(x - 3)

f(0) = c/-3 = 4

c = -12

f(x) = -12(x - 1)²/(x - 3)

Answer:

Step-by-step explanation:

The quadratic formula is y=ax^2+bx+c

If we move everything to the left side of the equation,

-6x^2=-9x+7 becomes

-6x^2+9x-7=0

a=-6, b=9, c=-7, so the third answer choice

       Hence,  The solution is:  

        x   =     1

        y   =  - 2

        z   =     2

Step-by-step explanation:

        3x  +  y  =  1

        x  -  y  +  z  =  5

        3x  +  y  +  z  =  3

        x  +  y  =  -1

        x  +  y  =   -1

        3x  +  y  =  1

        x   =   1

        y  =  -2

        1   -  (-2)  +   z    =   5

        z    =   2

       Hence, The Solution is:   x  =  1,   y  =  -2,    z   =  2

I hope this helps you!

We can conclude that there are no ordered triples of positive integers (x, y, z) that satisfy the given equation and the condition x < y.

We are given that the product of three positive integers (x, y, z) is equal to four times their sum:

xyz = 4(x + y + z)

Rearranging the equation, we get:

xyz - 4x - 4y - 4z = 0

We can factor out a common factor of 4 from the terms on the right-hand side:

4(xy - x - y - z) = 0

Now, we have two cases to consider:

Case 1: xy - x - y - z = 0

In this case, we can rewrite the equation as:

x(y - 1) - (y + z) = 0

From this equation, we observe that (y + z) must be divisible by (y - 1). Since x < y, the minimum value of (y - 1) is 1, which means (y + z) should also be 1. However, since we are looking for positive integers, this case does not yield any solutions.

Case 2: xy - x - y - z = 4

In this case, we can rewrite the equation as:

x(y - 1) - (y + z) = 4

Similarly, we observe that (y + z) must be divisible by (y - 1), and now (y - 1) can take on a minimum value of 2. We can analyze different possibilities based on this:

If (y - 1) = 2, then (y + z) = 2. Since we are dealing with positive integers, the only possibility is y = 3 and z = -1, which does not satisfy the condition.

If (y - 1) = 3, then (y + z) = 3. The only possibility is y = 4 and z = -1, which also does not satisfy the condition.

If (y - 1) = 4, then (y + z) = 4. The only possibility is y = 5 and z = -1, which does not satisfy the condition.

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Answer:The solution of the linear equations y = 4x − 19.4 and y = 0.2x − 4.2 will be (4, -3.4). Then the correct option is A.

What is the solution to the equation?

The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.

The equations are given below.

y = 4x − 19.4      ...1

y = 0.2x−4.2      ...2

From equations 1 and 2, then we have

4x - 19.4 = 0.2x - 4.2

3.8x = 15.2

x = 4

Then the value of the variable 'y' will be calculated as,

y = 4 (4) - 19.4

y = 16 - 19.4

y = - 3.4

The solution of the linear equations y = 4x − 19.4 and y = 0.2x − 4.2 will be (4, -3.4). Then the correct option is A.

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Answer:

1080yd³

Step-by-step explanation:

1/2x12x9=54

54x20=1080

The total volume of the dill spears is 425 cm³.

To find the total volume of the dill spears, we can subtract the volume of the pickle juice from the volume of the jar.

The jar is in the shape of a cylinder with a base area of 45 cm² and a height of 13 cm. Therefore, the volume of the jar can be calculated using the formula:

Volume of the jar = base area * height

Volume of the jar = 45 cm² * 13 cm

Volume of the jar = 585 cm³

Now, we know that the measuring cup collected 160 cm³ of pickle juice. So, we subtract this volume from the total volume of the jar to find the volume of the dill spears.

Volume of the dill spears = Volume of the jar - Volume of the pickle juice

Volume of the dill spears = 585 cm³ - 160 cm³

Volume of the dill spears = 425 cm³

Therefore, the total volume of the dill spears is 425 cm³.

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Answer:

  (c)  y = 2b/(2a -c)x -2bc/(2a -c)

Step-by-step explanation:

Given the equation of line BD in point-slope form you want the simplified equation.

  y -0 = (2b/(2a -c))(x -c)

The equation is simplified by using the distributive property to eliminate parentheses.

  [tex]y-0=\dfrac{2b}{2a-c}(x -c)\\\\\\y=\dfrac{2b}{2a-c}x-\dfrac{2b}{2a-c}c\\\\\\\boxed{y=\dfrac{2b}{2a-c}x-\dfrac{2bc}{2a-c}}\qquad\text{matches choice C}[/tex]

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In a right triangle, the length of side "a" is 12.

The Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, can be used to find the length of side "a" in a right triangle with sides of 13 and 5 units.

Let's assign "a" as the unknown side. According to the Pythagorean theorem, we have the equation: [tex]a^{2}[/tex] = [tex]13^{2}[/tex] - [tex]5^{2}[/tex].

Simplifying the equation, we get [tex]a^{2}[/tex] = 169 - 25, which becomes [tex]a^{2}[/tex] = 144.

To solve for "a," we take the square root of both sides: a = √144.

The square root of 144 is 12. Therefore, side "a" has a length of 12 units.

In summary, using the Pythagorean theorem, we determined that side "a" in the right triangle with side lengths 13 and 5 units has a length of 12 units.

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Work Shown:

cos(angle) = adjacent/hypotenuse

cos(S) = TS/SR

cos(S) = 63/65

cos(S) = 0.969231

cos(S) = 0.97

Each decimal value is approximate. See the diagram below.

Step-by-step explanation:

To find a system with the same solution as the given system, we can multiply both sides of both equations by a nonzero constant, which will result in a system that is equivalent to the original one.

For example, let's multiply the first equation by 2 and the second equation by 3:

First equation (multiplied by 2):

6x + 4y = -10

Second equation (multiplied by 3):

6x + 9y = 15

The new system of equations is:

6x + 4y = -10

6x + 9y = 15

This system has the same solution as the original system because it's just a scalar multiple of the original system.

The correct option that would be sufficient to prove the right triangles ∆WXZ and ∆WYX is (A) WZ/WX = XW/YW

The perpendicular height of the right triangle divides the triangle in two triangles with the same proportions as the original triangle.

Considering the smaller right triangle ∆WXZ and the bigger triangle ∆WYX;

the side WZ of ∆WXZ will correspond to the side WX of ∆WYX and similarly, side XW of ∆WXZ will correspond to the side of ∆WYX

so we can we the proportion as;

WZ/WX = XW/YW

Therefore, the proportion WZ/WX = XW/YW would be sufficient to prove the right triangles ∆WXZ and ∆WYX

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Answer:

Option (C), 8 am

Step-by-step explanation:

Newton's Law of Cooling is a mathematical model that describes the cooling process of an object. It states that the rate of change of temperature of an object is proportional to the difference between its temperature and the surrounding temperature.

The equation representing Newton's Law of Cooling is:

[tex]\dfrac{dT}{dt} = -k (T_0 - T_A)[/tex]

Where...

After solving the differential equation we get the following function:

[tex]T(t)=T_A+(T_0-T_A)e^{-kt}[/tex]

[tex]\hrulefill[/tex]

Given:

[tex]T_0=98.6 \ \textdegree F \ \text{(This is the average human body temperature)}\\\\T_f=T(t)=80\ \textdegree F \\\\T_A=40 \ \textdegree F \\\\k=0.1947[/tex]

Find:

[tex]T(??)= \ 80 \ \textdegree F[/tex]

Substituting the values into the formula:

[tex]T(t)=T_A+(T_0-T_A)e^{-kt}\\\\\\\Longrightarrow 80=40+(98.6-40)e^{-0.1947t}\\\\\\\Longrightarrow 80=40+58.6e^{-0.1947t}\\\\\\\Longrightarrow 40=58.6e^{-0.1947t}\\\\\\\Longrightarrow 0.682594=e^{-0.1947t}\\\\\\\Longrightarrow \ln(0.682594)=-0.1947t\\\\\\\Longrightarrow t=\dfrac{\ln(0.682594)}{-0.1947} \\\\\\\therefore \boxed{t \approx 2 \ \text{hours}}[/tex]

Thus, we can conclude the time of death was at 8 am.

Answer:

A+B+C=28

Step-by-step explanation:

let

A-B=2 -----1 EQUATION

B-C=7-------2

A+C=17------3

FROM 1 AND 2

A-C=9---------4

FROM 2 AND 3

A+B=24 -------5

FROM 3 AND 4

2A=26

A=13 SUBSTITUTING A=13 IN 5

WE GET B=11 SUBSTITUTING IT IN 2

WE GET C=4

NOW

A+B+C=13+11+4=28

The solution to the equation [tex]3^x + 3^(4-2x) = 1 + 3^(4-x) is x = 2.[/tex]

To solve the equation [tex]3^x + 3^(4-2x) = 1 + 3^(4-x),[/tex] we can simplify the equation and then apply some algebraic techniques to isolate the variable x.

First, let's simplify the equation step by step:

1. Notice that [tex]3^(4-2x)[/tex] can be rewritten as[tex](3^4) / (3^2x)[/tex], using the property of exponentiation.

2. Now the equation becomes 3[tex]^x + (81 / 9^x) = 1 + 3^(4-x).[/tex]

3. We can simplify further by multiplying both sides of the equation by 9^x to eliminate the denominators.

  This gives us [tex]3^x * 9^x + 81 = 9^x + 3^(4-x) * 9^x.[/tex]

4. Simplifying the terms, we have [tex](3*9)^x + 81 = 9^x + (3*9)^(4-x).[/tex]

  Now we have [tex](27)^x + 81 = 9^x + (27)^(4-x).[/tex]

5. Notice that [tex](27)^x and (27)^(4-x)[/tex] have the same base, so we can set the exponents equal to each other.

  This gives us x = 4 - x.

6. Simplifying the equation, we get 2x = 4.

7. Dividing both sides of the equation by 2, we have x = 2.

Therefore, the solution to the equation [tex]3^x + 3^(4-2x) = 1 + 3^(4-x) is x = 2.[/tex]

Using simple language, we simplified the equation step by step and isolated the variable x by setting the exponents equal to each other. The final solution is x = 2.

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Answer:

The parabola has a vertex at (3, -4), has a p-value of -6 and it opens downwards.

Step-by-step explanation:

The given directrix of the parabola is y = 2, which is a horizontal line.

This means that the parabola is vertical, with a vertical axis of symmetry.

The focus of a parabola is a fixed point located inside the curve. The y-coordinate of the given focus is y = -10. As this is below the directrix, it means that the parabola opens downwards.

The standard form of a vertical parabola is:

[tex]\boxed{(x-h)^2=4p(y-k)}[/tex]

where:

As the focus is (3, -10), then:

[tex](h, k+p)=(3,-10)[/tex]

     [tex]\implies h = 3[/tex]

[tex]\implies k+p=-10[/tex]

As the directrix is y = 2, then:

[tex]k - p=2[/tex]

To find the value of k, sum the equations involved k and p to eliminate p:

[tex]\begin{array}{crcccr}&k &+& p& =& -10\\+&k& -& p& = &2\\\cline{2-6}&2k&&& =& -8\\\cline{2-6}\\\implies &k&&&=&-4\end{array}[/tex]

To find the value of p, substitute the found value of k into one of the equations:

[tex]-4-p=2[/tex]

        [tex]p=-4-2[/tex]

        [tex]p=-6[/tex]

Therefore, the values of h, k and p are:

The parabola has a vertex at (3, -4), has a p-value of -6 and it opens downwards.

The parabola has a vertex at (3, -4), has a p-value of -6 and it opens downwards.

In Mathematics, the standard form of the equation of the directrix lines for any parabola is given by this mathematical equation:

(x - h)² = 4p(y - k).

Where:

Since the directrix is horizontal, the axis of symmetry would be vertical. This ultimately implies that, we would have the following parameters;

directrix is y = 2

Focus, (h, k + p) = (3, -10)

Next, we would determine the value of k as follows;

k + p = -10     .......equation 1

k - p = 2     .......equation 2

By solving the equations simultaneously, we have:

2k = -8

k = -4

For the value of p, we have the following from equation 2:

k - p = 2

-4 - p = 2

p = -4 - 2

p = -6

In conclusion, we can logically deduce that the parabola opens downward because the p-value is negative.

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Answer:

x<3

Step-by-step explanation:

[tex] - 2(5 - 4x) < 6x - 4 \\ - 10 + 8x < 6x - 4 \\ 8x - 6x < - 4 + 10 \\ 2x < 6 \\ x < 3[/tex]